Finite Group Databasebeta

← #73083

Subgroups — #73083

Order 384.

168 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#127⋊C3C211yesmaximal, characteristic
2#4623C212yesmaximal, complemented
3#1887—31nomaximal, Sylow
4C2×SL(2,ℤ/4)—41nomaximal
5#127⋊C3C412yescomplemented
6C2×C42⋊C3C2211yescharacteristic
7C2×#127D311yescharacteristic
8#518—31no—
9#418—32no—
10C2×#100—31no—
11#339—32no—
12C22×A4—41no—
13SL(2,ℤ/4)—42no—
14C42⋊C3C2×C411yescharacteristic, complemented
15C22×D4—31no—
16C22×D4—31no—
17C2×M16—32no—
18#127—32no—
19#127Dic312yescomplemented
20C4×D4—62no—
21C4×D4—62no—
22#117—32no—
23C2×C22⋊C4—61no—
24C2×C22⋊C4—61no—
25C2×C42C2×D311yescharacteristic
26#102—62no—
27#102—62no—
28#100—62no—
29C2×A4—42no—
30C2×A4—41no—
31C2×Dic3—161no—
32C24—31no—
33C24S411yescharacteristic
34C2×D4—32no—
35C2×D4—32no—
36C2×D4—32no—
37C2×D4—62no—
38C2×D4—62no—
39C2×D4—62no—
40C2×D4—62no—
41C2×D4—61no—
42C22×C4—31no—
43C22×C4—61no—
44C22×C4—61no—
45C22×C4—61no—
46C22×C4—61no—
47M16—62no—
48M16—62no—
49C2×C8—62no—
50C4⋊C4—32no—
51C4⋊C4—32no—
52C22⋊C4—62no—
53C22⋊C4—62no—
54C22⋊C4—62no—
55C42—31no—
56C42—62no—
57C42C2×Dic311yescharacteristic, complemented
58C22×C3—161no—
59A4—41no—
60Dic3—162no—
61C23—32no—
62C23—61no—
63C23—61no—
64C23—61no—
65C23—61no—
66C23—121no—
67C23—121no—
68C23SL(2,ℤ/4)12yes—
69C23C2×S411yescharacteristic
70D4—64no—
71D4—64no—
72D4—64no—
73D4—122no—
74C2×C4—32no—
75C2×C4—32no—
76C2×C4—31no—
77C2×C4—31no—
78C2×C4—62no—
79C2×C4—62no—
80C2×C4—241no—
81C2×C4—241no—
82C2×C4—241no—
83C2×C4—241no—
84C8—122no—
85C2×C3—162no—
86C2×C3—161no—
87C22—31no—
88C22—31no—
89C22—41no—
90C22—62no—
91C22—62no—
92C22—121no—
93C22—122no—
94C22—121no—
95C22—121no—
96C22—241no—
97C22C2×SL(2,ℤ/4)11yescharacteristic
98C4—32no—
99C4—32no—
100C4—122no—
101C4—122no—
102C3—161noSylow
103C2—31no—
104C2—31no—
105C2—42no—
106C2—122no—
107C2#857⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2#857⋊C2
commutator, hypocenterC42⋊C3C2×C4
Frattini, socleC23C2×S4
FittingC2×#127D3
prime radicalC2×#127⋊C3C2
cosocleC2×C42⋊C3C22
LCSG ▷ C42⋊C3
UCSC1 ◁ C2

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #4623C2semidirect
5. #127⋊C3C4semidirect
14. C42⋊C3C2×C4semidirect
19. #127Dic3semidirect
57. C42C2×Dic3semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#127⋊C32
#46232
#18873
C2×SL(2,ℤ/4)4